<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:58"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">L</font>]
 ;<br/>

<b>let </b><font color="Maroon">SL</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">SS</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>;<br/>





<a NAME="E2:58"/><i><font color="Green">E47</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SS</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">SS</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">SL</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font>;<br/>
<a NAME="E3:58"/><i><font color="Green">E48</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<a NAME="E4:58"/><b>then </b>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">SL</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:58"/><b>then </b>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">SL</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E6:58"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font> is <a href="waybel_0.html#V23">isomorphic</a> &amp; ( for <font color="Olive">W</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span> holds  <span class="p1">(<span class="default"><a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Olive">W</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="waybel26.html#K6">*graph</a>  ) )
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:58"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">f1</font> is <a href="waybel_0.html#V23">isomorphic</a> &amp; ( for <font color="Olive">f</font> being <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> holds  <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> ) )
 <b>by </b><i><a class="ref" href="waybel27.html#T33">WAYBEL27:33</a></i>;<br/>
<b>set </b><font color="Maroon">f2</font> = <font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> ;<br/>
<a NAME="E9:58"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, <a class="ref" href="yellow14.html#T11">YELLOW14:11</a></i>;<br/>
<b>consider </b><font color="Maroon">f3</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">L</font>,<span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:58"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">f3</font> is <a href="waybel27.html#V1">uncurrying</a> &amp; <font color="Maroon">f3</font> is <a href="waybel_0.html#V23">isomorphic</a> )
 <b>by </b><i><a class="ref" href="waybel27.html#T47">WAYBEL27:47</a></i>;<br/>
<b>set </b><font color="Maroon">f4</font> = <font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> ;<br/>
<a NAME="E12:58"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, <a class="ref" href="yellow14.html#T11">YELLOW14:11</a></i>;<br/>
<a NAME="E13:58"/>
 <a href="waybel27.html#K2">UPS</a> <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4">WAYBEL27:def 4</a></i>;<br/>
<a NAME="E14:58"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">L</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow16.html#T41">YELLOW16:41</a></i>;<br/>
<a NAME="E15:58"/>
 <a href="waybel27.html#K2">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">L</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">L</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4">WAYBEL27:def 4</a></i>;<br/>
<a NAME="E16:58"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="waybel27.html#K2">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">L</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="waybel15.html#T1">WAYBEL15:1</a></i>;<br/>
<a NAME="E17:58"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E18:58"/><i><font color="Green">E78</font></i>: 
 <a href="waybel27.html#K2">UPS</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4">WAYBEL27:def 4</a></i>;<br/>
<a NAME="E19:58"/>
( <font color="Maroon">f3</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f3</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a>, <a class="ref" href="yellow14.html#T10">YELLOW14:10</a></i>;<br/>
<a NAME="E20:58"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="waybel27.html#V2">currying</a>
 
<b>by </b><i>, , , <a class="ref" href="waybel29.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>consider </b><font color="Maroon">f5</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">L</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E22:58"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">f5</font> is <a href="waybel_0.html#V23">isomorphic</a> &amp; ( for <font color="Olive">f</font> being <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> holds  <font color="Maroon">f5</font> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> ) )
 <b>by </b><i><a class="ref" href="waybel27.html#T33">WAYBEL27:33</a></i>;<br/>
<b>set </b><font color="Maroon">f6</font> =  <a href="waybel27.html#K4">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f5</font>;<br/>
<a NAME="E23:58"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<a NAME="E24:58"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="waybel27.html#K4">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f5</font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, <a class="ref" href="waybel27.html#T35">WAYBEL27:35</a></i>;<br/>
<b>set </b><font color="Maroon">G</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4">UPS</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f5</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>;<br/>
<a NAME="E25:58"/>
<span class="p1">(<span class="default"><a href="waybel27.html#K4">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f5</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="ref" href="waybel29.html#T1" target="_self">Th1</a>, , </i>;<br/>
<a NAME="E26:58"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4">UPS</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f5</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E27:58"/><i><font color="Green">E119</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> #)
 
<b>by </b><i>, <a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E28:58"/><i><font color="Green">E121</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">SS</font>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">SS</font>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:58_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4">WAYBEL27:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">SS</font>,<span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span> <b>by </b><i>, <a class="txt" href="waybel29.html#E2:58"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="waybel27.html#D4">WAYBEL27:def 4</a></i>;<br/>
<a NAME="E4:58_1_1"/>
<font color="Maroon">x1</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i>, <a class="txt" href="waybel29.html#E2:58"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel29.html#E6:58"><i><font color="Green">E50</font></i></a>, <a class="ref" href="waybel21.html#T6">WAYBEL21:6</a></i>;<br/>
<a NAME="E5:58_1_1"/><b>then </b>
<font color="Maroon">x1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T22">WAYBEL17:22</a></i>;<br/>
<b>hence </b><a NAME="E6:58_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">SS</font>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel24.html#D3">WAYBEL24:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:58_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">SS</font>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">SS</font>,<span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:58_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x1</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="waybel24.html#D3">WAYBEL24:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i>, <a class="txt" href="waybel29.html#E2:58"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E6:58_1"/>
<font color="Maroon">x1</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i>, <a class="ref" href="waybel17.html#T22">WAYBEL17:22</a></i>;<br/>
<a NAME="E7:58_1"/><b>then </b>
<font color="Maroon">x2</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i>, <a class="txt" href="waybel29.html#E2:58"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel21.html#T6">WAYBEL21:6</a></i>;<br/>
<b>hence </b><a NAME="E8:58_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel27.html#D4">WAYBEL27:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4">UPS</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f5</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">SS</font>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span></span>)</span> ;<br/>
<a NAME="E30:58"/><i><font color="Green">E125</font></i>: 
for <font color="Olive">V</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span><br/> for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>



<b>reconsider </b><font color="Maroon">fff</font> = <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">L</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel27.html#D4">WAYBEL27:def 4</a></i>;<br/>
<a NAME="E2:58_2"/>
<font color="Maroon">f5</font> is <a href="waybel_0.html#V18">sups-preserving</a>
 
<b>by </b><i>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<a NAME="E3:58_2"/><b>then </b>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <font color="Maroon">L</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span>  st not <font color="Olive">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">X</font> is <a href="waybel_0.html#V1">directed</a> holds <br/><font color="Maroon">f5</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">X</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>
<a NAME="E4:58_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">f5</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E5:58_2"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f5</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">fff</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f5</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">fff</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>;<br/>
<a NAME="E6:58_2"/><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="waybel27.html#K3">.</a> <font color="Maroon">s</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4">WAYBEL27:def 4</a></i>;<br/>
<font color="Maroon">G</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4">UPS</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f5</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel27.html#K4">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f5</font></span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">f5</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">fff</font>
<b>by </b><i><a class="txt" href="waybel29.html#E6:58"><i><font color="Green">E50</font></i></a>, <a class="ref" href="waybel27.html#D5">WAYBEL27:def 5</a>, </i>
;<br/>
<b>then </b><i><font color="Green">E130</font></i>: <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> = 
<font color="Maroon">f5</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">fff</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="waybel27.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, </i>
;<br/>
<b>reconsider </b><font color="Maroon">f2V</font> = <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <b>by </b><i><a class="ref" href="waybel27.html#D4">WAYBEL27:def 4</a></i>;<br/>
<a NAME="E10:58_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2">UPS</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:58_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K3">curry</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel27.html#D2">WAYBEL27:def 2</a></i>;<br/>
<a NAME="E12:58_2"/>
<font color="Maroon">f1</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i>, <a class="ref" href="yellow14.html#T10">YELLOW14:10</a></i>;<br/>
<a NAME="E13:58_2"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E14:58_2"/><b>then </b><i><font color="Green">E133</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<i><font color="Green">E134</font></i>: <font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="waybel29.html#T6" target="_self">Th6</a>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">f2V</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i/>
;<br/>
<b>thus </b><a NAME="E16:58_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Olive">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:58_2_4"/><b>then </b><i><font color="Green">E135</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="waybel27.html#K3">.</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="waybel27.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E8:58"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:58_2_4"/><i><font color="Green">E136</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:58_2_4"/><b>then </b><i><font color="Green">E137</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span>
 
;<br/>
<a NAME="E6:58_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="waybel27.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="waybel27.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:58_2_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_5.html#T27">FUNCT_5:27</a>, </i>;<br/>
<a NAME="E8:58_2_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:58_2_4"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:58_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Olive">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E17:58_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Olive">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E19:58_2"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/>
<a NAME="E20:58_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:58_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">cs</font> = <span class="p1">(<span class="default"><a href="funct_5.html#K3">curry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="funct_5.html#T26">FUNCT_5:26</a></i>;<br/>
<a NAME="E23:58_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E24:58_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E25:58_2"/><b>then </b>
<font color="Maroon">cs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="funct_5.html#T27">FUNCT_5:27</a>, </i>;<br/>
<a NAME="E26:58_2"/><b>then </b><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="waybel27.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="waybel27.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E27:58_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f3</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="waybel27.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E28:58_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="waybel29.html#E8:58"><i><font color="Green">E53</font></i></a>, , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:58"/><i><font color="Green">E141</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_3"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span> ;<br/>
<a NAME="E3:58_3"/>
 <a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i>, <a class="ref" href="yellow14.html#T10">YELLOW14:10</a></i>;<br/>
<a NAME="E4:58_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">SS</font>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:58_3"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:58_3"/><i><font color="Green">E143</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:58_3"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W2</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E11:58_3"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_3_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:58_3_1"/>
<font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E3:58_3_1"/><b>then </b>
<font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SS</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">w1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">w2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:58_3_1"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">w2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w2</font> = <font color="Maroon">w2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:58_3_1"/>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">w2</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">w1</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="waybel29.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:58_3_1"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">w1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>
<a NAME="E10:58_3_1"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="waybel26.html#K6">*graph</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font>
 
<b>by </b><i><a class="ref" href="waybel26.html#D5">WAYBEL26:def 5</a></i>;<br/>
<a NAME="E11:58_3_1"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:58_3_1"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">w1</font>,<font color="Olive">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">w2'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:58_3_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">w2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">w2'</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/>
<b>thus </b><a NAME="E15:58_3_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel29.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:58_3"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_3_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E2:58_3_2"/>
<font color="Maroon">V1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:58_3_2"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:58_3_2"/><b>then </b>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
;<br/>
<a NAME="E5:58_3_2"/><b>then </b>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">v1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">v2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:58_3_2"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E10:58_3_2"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Olive">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:58_3_2"/><b>then </b>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:58_3_2"/><b>then </b>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="waybel26.html#K6">*graph</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:58_3_2"/><b>then </b>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel26.html#D5">WAYBEL26:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">v1'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:58_3_2"/><i><font color="Green">E153</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v1'</font>,<font color="Maroon">v2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">v1'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v1</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>
<b>thus </b><a NAME="E16:58_3_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel29.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:58_3"/><b>then </b>
<font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E8:58"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E14:58_3"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E32:58"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_4"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> ;<br/>
<a NAME="E3:58_4"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W2</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:58_4"/>
<font color="Maroon">G</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:58"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel29.html#E3:58"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow14.html#T10">YELLOW14:10</a></i>;<br/>
<a NAME="E6:58_4"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">SS</font>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:58_4"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">W2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E10:58_4"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_4_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E2:58_4_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:58_4_1"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:58_4_1"/><b>then </b>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
;<br/>
<a NAME="E5:58_4_1"/><b>then </b>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">v1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">v2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:58_4_1"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E10:58_4_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Olive">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:58_4_1"/><b>then </b>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:58_4_1"/><b>then </b>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="waybel26.html#K6">*graph</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:58_4_1"/><b>then </b>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel26.html#D5">WAYBEL26:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">v1'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:58_4_1"/><i><font color="Green">E159</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v1'</font>,<font color="Maroon">v2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">v1'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v1</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>
<b>thus </b><a NAME="E16:58_4_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel29.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:58_4"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_4_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:58_4_2"/>
<font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E3:58_4_2"/><b>then </b>
<font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SS</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">w1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">w2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:58_4_2"/><i><font color="Green">E245</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">w2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w2</font> = <font color="Maroon">w2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:58_4_2"/>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">w2</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">w1</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:58"><i><font color="Green">E54</font></i></a>, <a class="ref" href="waybel29.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:58_4_2"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">w1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>
<a NAME="E10:58_4_2"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="waybel26.html#K6">*graph</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font>
 
<b>by </b><i><a class="ref" href="waybel26.html#D5">WAYBEL26:def 5</a></i>;<br/>
<a NAME="E11:58_4_2"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel29.html#K3" target="_self">Theta</a> <font color="Maroon">SS</font>,<font color="Maroon">SL</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:58_4_2"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">w1</font>,<font color="Olive">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">w2'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:58_4_2"/><i><font color="Green">E246</font></i>: 
( <font color="Maroon">w2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">w2'</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/>
<b>thus </b><a NAME="E15:58_4_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="waybel29.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel29.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:58_4"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E8:58"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E13:58_4"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E33:58"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel29.html#E6:58"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b> <a href="waybel11.html#K5">sigma</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
<br/>.= 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
;<br/><br/>


</div>
